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Thread: factor higher order polynomials

  1. #1
    Super Member
    Jul 2010

    factor higher order polynomials

    I know how to dertermine if quadratics factor

    But how do you dertermine if higher order polynomials factor?

    Is it just trial and error and remainder theorem?

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  2. #2
    MHF Contributor
    Prove It's Avatar
    Aug 2008

    Re: factor higher order polynomials

    All cubic and quartic equations can be solved. Any higher polynomial can not.
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  3. #3
    MHF Contributor

    Apr 2005

    Re: factor higher order polynomials

    Well, let's call it educated trial and error! To determine whether or not a polynomial has a factor with integer coefficients we can use the "rational root theore": If $\displaystyle \frac{p}{q}$ satisfies the polynomial equation $\displaystyle a_nx^n+ a_{n-1}x^{n-1}+\cdot\cdot\cdot+ a_1x+ a_0= 0$ then the numerator, p, evenly divides the "constant term", $\displaystyle a_0$, and the denominator, m, evenly divides the "leading coefficient", $\displaystyle a_n$. That allows you to reduce to a finite number of trials. However, Most polynomials do NOT factor with integer or rational number coefficients. There is no good "trial and error" method to find non-rational factors or non-rational roots except to use "completing the square" for quadratics and the general formulas for quadratic, cubic, and quartic polynomials. There are no formulas for higher degree equations in terms or radicals because some higher degree equations have roots that cannot be written in terms of radicals.
    Last edited by HallsofIvy; Aug 3rd 2012 at 04:58 AM.
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